Asymptotic Behavior of Weak Solutions to the Generalized Nonlinear Partial Differential Equation Model

(−Δ) 1/2 is the fractional power of the Laplacian Δ, and f(x, t) is an external force. The model is relevant to the theory of the atmosphere and ocean dynamics (refer to [1–4] and references therein). It should be mentioned that there are many results on the stability behaviors of the atmosphere and ocean dynamics in which the derivation is mainly based on the linear and nonlinear stability together with the numerical simulation [5–7]. Recently, Hu [8] investigated the following semilinear parabolic partial differential equation with Laplacian in R:


Introduction
In the past ten years, the study of the fractional order differential equation has attracted more and more attention.In this study, we consider a class of two-dimensional fractional order generalized nonlinear partial differential equation model which is governed by the differential equation together with the initial condition  (, 0) =  0 () .
The model is relevant to the theory of the atmosphere and ocean dynamics (refer to [1][2][3][4] and references therein).It should be mentioned that there are many results on the stability behaviors of the atmosphere and ocean dynamics in which the derivation is mainly based on the linear and nonlinear stability together with the numerical simulation [5][6][7].Recently, Hu [8] investigated the following semilinear parabolic partial differential equation with Laplacian in R 3 : and derived the error estimates between the solution of semilinear parabolic partial differential equation ( 3) and the solution of the linear heat equation.However, it is a challenging problem to consider the fractional order partial differential equation due to some new difficulty.One may also refer to some interesting and important results on the stability of the nonlinear partial differential equations [9][10][11].
In this study, we will investigate the asymptotic stability for solution of the two-dimensional fractional order partial differential equation (1) under the finite energy initial data  0 .
To do so, we first consider the perturbed fractional order partial differential equation: Here,  0 is any initial perturbation which may be large.We will show that every perturbed solution V of the fractional order partial differential equation (4) asymptotically converges to that of fractional order partial differential equations (1)- (2).That is to say, We now give the definition of solution for fractional order partial differential equations ( 1)-(2) (refer to [12]).
if the conditions (iii) energy inequality are valid.
Our result now reads.
Remark 4. Our methods are mainly based on the generalized Fourier splitting methods which are first used by Schonbek [13] (see also [14][15][16]) on the time decay issue of the classic Navier-Stokes equations and related partial differential equations [17].

Preliminaries
In this study, we denote by 's the abstractly positive constants which may be different from line to line.We denoted by   (R 2 ) the usual Lebesgue space with the norm We also denoted by Ḣ (R 2 ) the homogeneous fractional Sobolev space: Here, φ is Fourier transformation: In order to prove our main result, we now give some important lemmas which play a central role in the argument of the next section.
Proof of Lemma 6.Since  satisfies ess sup applying Gagliardo-Nirenberg inequality in Lemma 6, the direct computation becomes where for where we used the following relation: That is to say, In particular, we take  = ; then Then, Hence, we complete the proof of Lemma 6.

Stability of the Solution
We now prove Theorem 2. Firstly, as stated in the proof of Lemma 7, since (, ), V(, ) are two solutions of ( 1)-( 2) and ( 4), respectively, we take  =  − V; then Taking the  2 inner product of (36), it follows that For the right hand side of above equation, Inserting the above inequality into the right hand side of (37), it follows that (−Δ)